We propose a new reformulation of the cubic regularization subproblem. The reformulation is an unconstrained convex problem that requires computing the minimum eigenvalue of the Hessian. Then, based on this reformulation, we derive a variant of the (non-adaptive) CR provided a known Lipschitz ...
corresponds to newton-like directions/algorithms, yielding adaptive cubic regularization (ar2/arc) variants [ 14 ]. in this paper, our primary focus is on the case when \(p =3\) . this construction, as given in (ar p model), corresponds to the subproblem in the adaptive regularization ...
method [23], the stochastic adaptive regularization methods using cubics (SARC) [24–26] are proposed to address relatively small-scale nonconvex stochastic optimization problems, and they find the minimizer of a local second order Taylor approximation with a cubic regularization term at each ...
Complexity analysisThe cubic regularization(CR)algorithm has attracted a lot of attentions in the literature in recent years.We propose a new reformulation of the cubic regularization subproblem.The reformulation is an unconstrained convex problem that requires computing the minimum eigenvalue of the ...
At each iteration a candidate search direction is determined by solving the affine scaling cubic regularization subproblem and the new iteration is strictly feasible by way of an interior backtracking technique. The global convergence and local superlinear convergence of the proposed algorithm are ...
The adaptive cubic regularization algorithms described in Cartis, Gould and Toint [Adaptive cubic regularisation methods for unconstrained optimization Part II: Worst-case function- and derivative-evaluation complexity, Math. Program. (2010), doi:10.1007/s10107-009-0337-y (online)]; [Part I: ...
Cubic regularizationWorst-case complexityMatrix-free subproblem solversIn this paper we consider the problem of minimizing a smooth function by using the adaptive cubic regularized (ARC) framework. We focus on the computation of the trial step as a suitable approximate minimizer of the cubic model ...
We propose a first-order method to solve the cubic regularization subproblem (CRS) based on a novel reformulation. The reformulation is a constrained convex optimization problem whose feasible region admits an easily computable projection. Our reformulation requires computing the minimum eigenvalue of ...
Gratton. On the use of the energy norm in trust-region and adaptive cubic regularization subproblems, April 2017.E. Bergou, Y. Diouane, and S. Gratton. On the use of the energy norm in trust-region and adaptive cubic regularization subproblems. Technical report, 2017....
The latter solves the subproblem in nested Krylov subspaces by a Lanczos-based method, which requires the storage of a dense matrix that can be comparable to or larger than the two dense arrays required by our approach if the problem is large or requires many Lanczos iterations. Finally, we ...